Fund Project:
The research for this paper was supported by the Icelandic Research Fund (Rannís) in the project `Lyapunov Methods and Stochastic Stability' (152429-051), which is gratefully acknowledged

The γ-basin of attraction of the zero solution of a nonlinear stochastic differential equation can be determined through a pair of a local and a non-local Lyapunov function. In this paper, we construct a non-local Lyapunov function by solving a second-order PDE using meshless collocation. We provide a-posteriori error estimates which guarantee that the constructed function is indeed a non-local Lyapunov function. Combining this method with the computation of a local Lyapunov function for the linearisation around an equilibrium of the stochastic differential equation in question, a problem which is much more manageable than computing a Lyapunov function in a large area containing the equilibrium, we provide a rigorous estimate of the stochastic γ-basin of attraction of the equilibrium.